as an upper base construct a prism with lateral edges parallel to OC and altitude equal to h.

Each prism of O-ABC is equivalent to the prism next above it in O'-A'B'C', hence the difference between the two sets of prisms is the lowest prism of the first set. By increasing the number of parts into which each altitude is divided, each part hand hence the volume of the lowest prism in O'-A'B'C', can be diminished indefinitely. Or if P' and P denote the volumes of the two sets of prisms, the limit of P' - P = 0.

But

P'>V and V>P.

.. P-P>V' – V.

(Ax. 9.)

If, however, V' was greater than V, V'-V would be a positive quantity, or a positive quantity would be less than a quantity which approaches zero as a limit, which is impossible.

.. V' is not greater than V. In the same manner we may prove that pyramid Vis not greater than pyramid V'.

PROPOSITION XVII. THEOREM

593. The volume of a triangular pyramid is equal to one third of the product of its base by its altitude.

Hyp. Vis the volume, B the base, and a the altitude of the triangular pyramid E-ABC.

Proof. On ABC construct the prism ABC-DEF, having its lateral edges equal and parallel to EB.

The prism is composed of the pyramid E-ABC and the quadrangular pyramid E-ACFD.

Through ED and EC pass a plane cutting ACFD in DC, forming two triangular pyramids, E-DCF and E-ADC.

E-DCF E-ADC.

(Same altitude and equivalent bases.)

(1)

Pyramid E-ADC is the pyramid C-ADE, since any face of a tetraedron may be taken as base,

594. COR. 1. The volume of any pyramid is equal to one

third the product of its base by its altitude.

For the pyramid may be divided into triangular pyramids by passing planes through any edge and the corresponding diagonals of the base.

The bases of the triangular pyramids together form the base of the given pyramid, and the sum of the volumes of the triangular pyramids equals the volume of the given pyra

mid.

Therefore the volume of any pyra

D

E

M

mid equals one-third the product of its base by its altitude.

595. COR. 2. The volumes of two pyramids are to each other as the products of their bases and altitudes.

596. COR. 3. Pyramids having equivalent bases are to each other as their altitudes, and those having equal altitudes are tʊ each other as their bases.

597. COR. 4. Pyramids having equal altitudes and equivalent bases are equivalent.

598. SCHOLIUM. The volume of any polyedron may be obtained by dividing it into pyramids and finding their volumes separately.

PROPOSITION XVIII. THEOREM

599. The frustum of a triangular pyramid is equivalent to the sum of three pyramids each having the same altitude as the frustum and whose bases are the upper and lower bases of the frustum, and the mean proportional between these bases.

Hyp. ABC-DEF is a frustum of any triangular pyramid, with bases ABC [B], and DEF [b], and altitude H.

To prove ABC-DEF equivalent to the sum of three pyramids whose common altitude is H, and whose bases are ABC and DEF, and a mean proportional between them.

Proof. Through the vertices C, D, B, and C, D, E, pass planes, dividing the frustum into three pyramids.

The pyramids D-CAB and C-FDE have the common altitude H, and for bases respectively ABC and DEF, the lower and upper bases of the frustum.

It remains to show that C-DEB is equivalent to a pyramid whose altitude is H, and whose base is a mean proportional between ABC and DEF [√B × b].

Pyramids C-ABD and C-BDE have their bases in the plane ABED.

Hence,

.. C-ABD: C-BDE = ▲ ABD : ▲ BDE. (Why?) ▲ ABD: ▲ BDE = AB: DE.

C-ABD: C-BDE = AB: DE. (1)

(Why?)

(Why?)

The pyramid C-DBE may be read D-BCE, and we have the pyramids D-CEF and D-BCE with bases in the plane FEBC and vertices at D.

.. D-BCE: D-CEF::▲ BCE: A CEF.

But

(2)

=

C-ABD=Hx B, and D-CEF=Hxb. (Why?)

.. C-BED = √ }H × B × ¦ H × b = { H √ B × b.

Hence the frustum of the pyramid is equivalent to the sum of three pyramids, whose bases are the upper and lower bases of the frustum, and the mean proportional between these bases, and having the altitude, H.

600. COR. 1. Since the volume of a pyramid is equal to the base multiplied by one-third of its altitude, we have Vol. of frustum of a triangular pyramid = H(B+b+√Bxb).

601. COR. 2. The volume of the frustum of any pyramid is

equal to one-third of its altitude multiplied by the sum of its

bases and a mean proportional between them.

い

Ex. 1109. Find the volume of a pyramid whose base is 60, and whose altitude is 5.

Ex. 1110. Find the altitude of a pyramid whose volume is 200, and whose base is a rectangle having the sides 5 and 12 respectively.

Ex. 1111. The three sides of the base of a pyramid are respectively 10, 17, and 21. Find the volume if the altitude is 5.

Ex. 1112. The three sides of the base of a pyramid are respectively 9, 10, 17. Find the volume if a lateral edge is 20, and its projection upon the base equals 12.

Ex. 1113. A lateral edge of a pyramid equals 10, and its inclination to the base is 30°. Find the area of the base if the volume of the pyramid is 100.

Ex. 1114. The base of a pyramid is a rhombus whose diagonals are respectively 10 and 12. Find the volume if the altitude is 6.

Ex. 1115. The diagonals of a parallelopiped divide the figure into six equivalent pyramids.

Ex. 1116. If any point within a parallelopiped be joined to the 8 vertices, 6 pyramids are formed, of which the sum of any opposite two is equal to the sum of any other opposite two.

Ex. 1117. Each edge of a triangular pyramid is equal to 10. Find the volume.

Ex. 1118. The perimeter of the triangular base of a regular pyramid is 40. Find the volume if the altitude is 12.

Ex. 1119. The base of a pyramid is a parallelogram of base 10 and altitude 8. Find the volume if a lateral edge is equal to 6, and forms with the base an angle of 45°.

Ex. 1120. The base of a pyramid is a rectangle having sides respectively equal to a and b. A lateral edge is equal to c, and is inclined to the base 30°. Find the volume.

Ex. 1121. Find the altitude of a pyramid of base b, equivalent to another pyramid of base a and altitude h.

Ex. 1122. Find the volume of the frustum of a triangular pyramid, if the lower base is 9, the upper 4, and the altitude 5.